Question

If A is a square matrix of order 3, then $|(A-A^T{)}^{105}|$ is equal to

• A. $105|A|$
• B. $105|A{|}^2$
• C. $105$
• D. none of these

Answer 1

The correct option is D none of these

$\left|{(A-A^T)}^{105}\right|={|-(A^T-A)|}^{105}$

$=-{|A^T-A|}^{105}$
$=-{\left|{(A^T-A)}^T\right|}^{105}$ ($\because$ determinant of matrix and its transpose is equal)
$=-{|A-A^T|}^{105}$
$\Rightarrow {|A-A^T|}^{105}=-{|A-A^T|}^{105}=0(\because$ determinant of skew-symmetric matrix of odd order is 0.)

$\left|{(A-A^T)}^{105}\right|=0$
Hence, option 'D' is correct.